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Re: [PrimeNumbers] Re: Prime number "spans"

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  • Peter Kosinar
    ... If we let span(x) to be defined for all natural numbers as the smaller of the gaps between this number and the previous or next prime, it d be trivial to
    Message 1 of 8 , Dec 29, 2009
      > > This notion is derived from the familiar prime gaps: The "span" of a
      > > prime is the smaller of the two gaps adjacent to it. For example,
      > > span(839) is min(839-829-1,853-839-1) = min(9,13) = 9. One question
      > > that arises immediately: can one easily prove the existence of
      > > arbitrarily long spans (easy for gaps, using the factorial
      > > construction)? With only a casual look, a method is not obvious.
      > > Another question: are there any known tables of maximal spans?
      >
      > The factorial construction of proving a long gap is symmetric.
      >
      > If p is prime, then p# + 2, P# + 3, ...... up to p# + (p+1) are
      > necessarily composite. and also, p# -2, p#-3, p#-4, .... down to
      > p#-(p+1) are necessarily composite.
      >
      > So to prove arbitrarily long "spans", you need only prove that there are
      > infinitely many composites of the form (p# +1) or (p# - 1).

      If we let span(x) to be defined for all natural numbers as the smaller of
      the "gaps" between this number and the previous or next prime, it'd be
      trivial to show that it can attain arbitrarily large values. It can be
      done unconditionally using the factorial construction --- since the number
      x=((2n+2)! + n + 2) has at least "n" composites on each side, span(x) must
      be at least "n". For example, for n=3, composites 8!+2, 8!+3, 8!+4 precede
      8!+5 and 8!+6, 8!+7, 8!+8 follow it.

      The problem is to show that it can grow indefinitely even if we restrict
      ourselves to prime values of "x".

      Peter

      --
      [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
    • Kermit Rose
      ... Hello Peter. Thank you for this clarification. Kermit.
      Message 2 of 8 , Dec 29, 2009
        Peter Kosinar wrote:
        >
        >
        > The problem is to show that it can grow indefinitely even if we
        > restrict ourselves to prime values of "x".
        >
        > Peter
        >


        Hello Peter.

        Thank you for this clarification.

        Kermit.
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